Confidence Intervals
Our goals for today will be to estimate the population mean µfrom data when σis known, and
to estimate pfrom data for a binomial distribution. We do this by calculating something called a
confidence interval.
Alevel cconfidence interval for a parameter is an interval computed from data such that the
probability of the parameter being in that interval is c. The most common value of cis 0.95. Each
confidence interval will have the form “point estimate” ±“Margin of Error”.
For example, a 95% confidence interval for µwill be a range of values such that we are 95% certain
that µis in that interval.
1 Estimating the Population Mean, µ
Assume the following:
1. A simple random sample of size nwas taken from the population.
2. The value of σis known.
3. If the distribution of the data is normal, the sample size ncan be any number. If the distri-
bution is unknown, then we will require n≥30 (or higher in cases of distinct skew) in order
to apply the CLT.
The point estimate for µis ¯x. The margin of error is zc·σ
√n, where zcis a z-score such that
P(−zc< z < zc) = c. (Some of these values for zcare given on the bottom of page A16 in Appendix
B of your book.)
Therefore, the level cconfidence interval for µis
¯x±zc·σ
√n
Example
Julia has been jogging over a period of several years. The standard deviation of her times to run
2 miles is σ= 1.80. A random sample of 90 of her jogging times over the last year gives a sample
mean of 15.60 minutes. Find a 0.95 confidence interval for µand interpret your answer.
We are given σ= 1.80, ¯x= 15.60, and n= 90. z.95 = 1.96 from Table 4, so the confidence interval
is 15.60 ±1.96 ·1.80
√n= 15.60 ±0.37. This means that we are 95% certain that the population mean
for Julia’s run times is between 15.23 and 15.97.
2 Estimating pfor a Binomial Distribution
In order to use this technique, we will need to use data with at least 5 successes and at least 5
failures.
If our data set has sample size nand xsuccesses, our point estimate will be ˆp=x
n. Denote ˆq= 1 −ˆp.
The margin of error is zc·qˆpˆq
n, so the level cconfidence interval for pis
ˆp±zc·rˆpˆq
n= ˆp±zc·rˆp(1 −ˆp)
n
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